... S r = X 1 ···X r ,are all symmetric. These particular polynomials are called the elementary symmetricpolynomials.Theorem 1. 26 (Symmetric function theorem). Let A be a ring. Every symmetricpolynomial ... isomorphic to a ringed space of this form. A morphism of affine algebraic varieties is a morphism of ringed spaces; we often call it a regular map V → W ora morphism V → W,andwewriteMor(V,W) for the set ... in A /m mis m/ m m ; in particular, A /m mis a local ring. As s is not in m/ m m , it is a unit in A /m m,andsosa =0inA /m mimplies a =0inA /m m, i.e., a ∈ m m.We now prove that the map is surjective....